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A generalization of the Cassini formula

Alexey Stakhov

Doctor of Sciences in Computer Science, Professor

The International Club of the Golden Section

6 McCreary Trail, Bolton, ON, L7E 2C8, Canadagoldenmuseum@rogers.comwww.goldenmuseum.com

1. Introduction: Cassini formula

Great astronomer Giovanni Domenico Cassini. Cassini is the name of a famousdynasty of French astronomers. Giovanni Domenico Cassini (1625-1712) is the most

famous of them and the founder of this dynasty. The following facts illustrate his

outstanding contribution to astronomy. The name Cassini was given to numerousastronomical objects, the Cassini Crater on the Moon, the Cassini Crater on Mars,

Cassini Slot in Saturn's ring, and Cassini Laws of the Moons movement.However, the name of Cassini is widely known not only in astronomy, but also in

mathematics. Cassini developed a theory of the remarkable geometrical figures known

under the name ofCassini ovals. The mathematical identity that connects three adjacent

Fibonacci numbers is well known under the name Cassini formula.

Cassini formula for the Fibonacci numbers. The history of science is silent asto why Cassini took such a great interest in Fibonacci numbers. Most likely it was simplya hobby of the Great astronomer. At that time many serious scientists took a great interest

in Fibonacci numbers and the golden mean. These mathematical objects were also a

hobby of Cassinis contemporary, Kepler.

Now, let us consider the Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21, 34, . Take theFibonacci number 5 and square it, that is, 5

2=25. Multiply two Fibonacci numbers 3 and

8 that encircle the Fibonacci number 5, so we get 38=24.Then we can write:

52-38=1

Note that the difference is equal to (+1).

Now, we follow the same process with the next Fibonacci number 8, that is, at

first, we square it 82=64 and then multiply two Fibonacci numbers 5 and 13 that encircle

the Fibonacci number 8, 513=65. After a comparison of the result 513=65 with thesquare 8

2=64 we can write:

82-513=-1

Note that the difference is equal to (-1).

Further we have:

132-821=1,

212-1334=-1,

and so on.

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We see, that the quadrate of any Fibonacci number Fn always differs from the

product of the two adjacent Fibonacci numbers Fn-1 and Fn+1, which encircle it, by 1.However, the sign of 1 depends on the index nof the Fibonacci numberFn. If the index n

is even, then the number 1 is taken with minus, and if odd, with plus. The indicated

property of Fibonacci numbers can be expressed by the following mathematical formula:

21 1 ( 1)nn n nF F F

1+ + = . (1)

This wonderful formula evokes a reverent thrill, if one recognizes that this

formula is valid for any value ofn (we remember that n can be some integer in limits

from - up to +). The alternation of +1 and -1 in the expression (1) at the successivepassing of all Fibonacci numbers produces genuine aesthetic enjoyment and a feeling of

rhythm and harmony.

For centuries, it was believed that the Fibonacci numbers are the only number

sequence that has a unique mathematical property given by the Cassini formula (1).

In the late 20 th and early 21 th centuries, several researchers from different

countries the Argentinean mathematician Vera Spinadel [1], the French mathematicianMidhat Ghazal [2], the American mathematician Jay Kappraff [3], the Russian engineer

Alexander Tatarenko [4], the Armenian philosopher and physicist Hrant Arakelyan [5],

the Russian researcher Victor Shenyagin [6], the Ukrainian physicist Nikolai Kosinov

[7], Ukrainian-Canadian mathematician Alexey Stakhov [8,9], the Spanishmathematicians Falcon Sergio and Plaza Angel [10] and others independently began to

study a new class of recursive numerical sequences, which are a generalization of the

classical Fibonacci numbers. These numerical sequences led to the discovery of a newclass of mathematical constants, called Vera Spinadel "metallic proportions". [1].

The interest of a large number of researchers from different countries (USA,

Canada, Argentina, France, Spain, Russia, Armenia, Ukraine) can not be accidental. Thismeans that the problem of the generalized Fibonacci numbers has matured in modern

science.As shown in the works [1-9], the most attention of the researchers was drawn to

the "metallic means," while the properties of the generalized Fibonacci numbers s are

studied not enough. This article is an attempt to draw attention to some of the unusual

properties of generalized Fibonacci numbers, in particular, to the generalized Cassiniformula. For the first time, these results have been published in the author's article [10],

written in Russian.

2. The Fibonacci-numbers

Let us give a real number 0 > and consider the following recurrence relation:( ) ( ) ( )2 1F n F n F + = + + n ; ( ) ( )0 0, 1 1F F .= = (2)

The recurrence relation (2) "generates" an infinite number of new numerical

sequences, because every real number generates its own numerical sequence.Let us consider partial cases of the recurrence relation (2). For the case the

recurrence relation (2) is reduced to the following recurrence relation:

1 =

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( ) ( ) ( )2 1F n F n F n1 ( ) ( )1 1+ = + + ; 0 0, 1 1F F1 1 = , (3)=

This recurrence relation generates the classical Fibonacci numbers:

0,1,1, 2,3,5,8,13, 21,34,... . (4)

Based on this fact, we will name the numerical sequences, generated by the recurrence

relation (2), the Fibonacci -numbers.

For the case =2 the recurrence relation (2) is reduced to the recurrence relation

( ) ( ) ( )2 1F n F n F n2 ( ) ( )2 2+ = 2 + + ; 0 0, 1 1F F2 2 = , (5)=

which gives the so-called Pelly numbers

0,1,2,5,12,29,70,... (6)

For the cases the recurrence relation (2) is reduced to the following

recurrence relations:

3,4 =

( ) ( ) ( )2 1F n F n F n2 ( ) ( )2 2+ = 3 + + ; 0 0, 1 1F F2 2 = (7)=

( ) ( ) ( )2 1F n F n F n2 ( ) ( )2 2+ = 4 + + ; 0 0, 1 1F F2 2 = . (8)=

The -Fibonacci numbers have many remarkable properties, similar to the

properties of the classical Fibonacci numbers. It is proved that the Fibonacci -numbers,as well as the classical Fibonacci numbers can be "expanded" to the negative values of

the discrete variable n.

Table 1 shows the four expanded sequences of the Fibonacci -numbers,corresponding to the values .1,2,3,4 =

Table 1. The expanded - Fibonacci -numbers ( 1,2,3,4 = )

( )

( )

( )

( )

( )

( )

( )

( )

1

1

2

2

3

3

4

4

0 1 2 3 4 5 6 7 8

0 1 1 2 3 5 8 13 21

0 1 1 2 3 5 8 13 21

0 1 2 5 12 29 70 169 408

0 1 2 5 12 29 70 169 408

0 1 3 10 33 109 360 1189 3927

0 1 3 10 33 109 360 1199 3927

0 1 4 17 72 305 1292 5473 23184

0 1 4 17 72 305 1292 5473 23184

n

F n

F n

F n

F n

F n

F n

F n

F n

3. Cassini formula for the Fibonacci -numbersLet us prove that the Cassini formula (1) can be generalized for the case of the

Fibonacci -numbers. The generalized Cassini formula is of the following form:

( ) ( ) ( ) ( )12 1 1 1

n

F n F n F n+

+ = (9)

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Let us prove a validity of the formula (9) by using the induction by n. For the case

1n= the Fibonacci -numbers ( ) ( ) ( )1 , , 1F n F n F n + in the formula (9), according

to the recurrence relation (2), takes the following values:

( ) ( ) ( )0 0, 1 1, 2F F F = = = ,

what implies that the identity (9) for the case 1n= is equal:

( ) ( )21 0 1 =2

1

. (10)

The base of the induction is proved.

We formulate the following inductive assumption. Suppose that the identity (9) is

valid for any given integer n, and prove the validity of the identity (9) for the case of

1n+ , that is, the identity

( ) ( ) ( ) ( )22 1 2

n

F n F n F n+

+ + = (11)

is valid too.

In order to prove the identity (11), we represent the left part of the identity (11) asfollows:

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

2 2

2 2

2

2

1 2 1

1 1

1 1

1 1

F n F n F n F n F n F n F n

F n F n F n F n

F n F n F n F n

F n F n F n

1 + + = + + + =

+ +

= + +

= +

(12)

On the other hand, by using (9), we can rewrite:

( ) ( ) ( ) ( )1

21 1 1n

F n F n F n+

+ = . (13)

This means that the identity (12) can be rewritten as follows:

( ) ( ) ( ) ( ) ( )1 22 1 2 1

n n

F n F n F n 1+ +

+ + = =

The identity (11) is proved.

4. Numerical examples

Consider the examples of the validity of the identity (9) for the various sequences

shown in Table 1. Let us consider the ( )2F n sequence for the case ofn=7. For this case

we should consider the following triple of the Fibonacci 2-numbers ( )2F n :

. By performing calculations over them according

to (9), we obtain the following result:

( ) ( ) ( )2 2 26 70, 7 169, 8 408F F F= = =

( )2

169 70 408 28561 28560 1 = = ,

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what corresponds to the identity (9), because for the case n=7 we have:

.( ) ( )1 8

1 1n+

= = 1

Now let us consider the ( )3F n sequence from Table 1 for the case of . For

this case we should chose the following triple of the Fibonacci 3-numbers

6n=

( )3F n :

( ) ( ) ( )3 3 35 109, 6 360, 7 1189F F F= = = .

By performing calculations over them according to (9), we obtain the following result:

( )2

360 109 1189 129600 129601 1 = = ,

what corresponds to the identity (9), because for the case n=6 we have

.( ) ( )1 7

1 1n+

= = 1

Finally, let us consider the ( )4F n sequence from Table 1 for the case of

. For this case we should chose the following triple of the Fibonacci 4-numbers

::

5n=

( )4F n( ) ( ) ( )

4 4 44 72, 5 305, 6 1292F F F = = = .

By performing calculations over them according to (9), we obtain the following result:

( ) ( ) ( )2

305 72 1292 93025 93024 1 = = ,

what corresponds to the identity (9), because for the case n=-5 we have

.( ) ( )1 4

1 1n+

= = 1

Thus, by studying the generalized formula Cassini (9) for the Fibonacci -numbers, we came to the discovery of an infinite number of integer recurrence sequences

in the range from + to , with the following unique mathematical property,expressed by the generalized Cassini formula (9), which sounds as follows:

The quadrate of any Fibonacci -number ( )F n are always different from the

product of the two adjacent Fibonacci -numbers ( )1F n and ,( 1F n + ) which

surround the initial Fibonacci -number ( )F n , by the number 1; herewith the sign of the

difference of 1 depends on the parity ofn: ifn is even, then the difference of 1 is takenwith the sign minus, otherwise, with the sign plus.

Until now, we have assumed that only the classical Fibonacci numbers are of the

unusual property, given by the Cassini formula (1). However, as is shown above, a

number of such numerical sequences is infinite. All the Fibonacci -numbers given by (2)are of a similar property, given by the generalized Cassini formula (9)!

References

1. Vera W. de Spinadel. From the Golden Mean to Chaos. Nueva Libreria, 1998

(second edition, Nobuko, 2004).

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2. Midhat J. Gazale. Gnomon. From Pharaohs to Fractals. Princeton, New Jersey,

Princeton University Press -1999.- 280 p. (published in Russian, Moscow-

: Institute for the Computing Studies, 2002).

3. Kappraff Jay. Connections. The geometric bridge between Art and Science.

Second Edition. Singapore, New Jersey, London, Hong Kong. World Scientific,

2001.4. Tatarenko AA. The golden -harmonies and -fractals // Academy of

Trinitarism. Moscow: 77-6567, Electronic publication 12691, 09.12.2005 (in

Russian),

mT

mD

http://www.trinitas.ru/rus/doc/0232/009a/02320010.htm5. Arakelyan Hrant. The numbers and magnitudes in modern physics. Yerevan,

Publishing House. Armenian Academy of Sciences, 1989 (in Russian).

6. Shenyagin VP "Pythagoras, or how everyone creates his own myth." The fourteenyears after the first publication of the quadratic mantissas proportions.

// Academy of Trinitarism. Moscow: 77-6567, electronic publication 17031,

27.11.2011 (in Russian) http://www.trinitas.ru/rus/doc/0232/013a/02322050.htm7. Kosinov NV. Golden ratio, golden constants, and golden theorems. // Academy of

Trinitarism. Moscow: 77-6567, electronic publication 14379, 02.05.2007 (inRussian) http://www.trinitas.ru/rus/doc/0232/009a/02321049.htm

8. Stakhov AP. Gazale formulas, a new class of hyperbolic Fibonacci and Lucas

functions and the improved method of the "golden" cryptography. // Academy of

Trinitarism., oscow: 77-6567, Electronic publication 14098, 21.12.2006

http://www.trinitas.ru/rus/doc/0232/004a/02321063.htm9. Stakhov A.P. The Mathematics of Harmony. From Euclid to Contemporary

Mathematics and Computer Science. New Jersey, London, Singapore, Hong

Kong: World Scientific, 2009. 748 p.10.Falcon Sergio, Plaza Angel. On the Fibonacci k-numbers Chaos, Solitons &

Fractals, Volume 32, Issue 5, June 2007 : 1615-1624.

11.Stakhov AP. A theory of the Fibonacci -numbers // Academy of Trinitarism.Moscow: 77-6567, Electronic publication 17407, 05.04.2012 (in Russian)

http://www.trinitas.ru/rus/doc/0232/009a/02321250.htm

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